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%I #10 Dec 23 2024 02:03:05
%S 2,3,5,7,9,10,13,15,17,18,22,23,26,27,31,32,40,42,43,44,52,53,67,68,
%T 69,70,77,78,85,86,90,91,116,117,119,120,135,136,151,152,169,170,186,
%U 187,197,198,243,244,246,247,291,292,312,313,339,340,358,360,362
%N Numbers k such that there is a unique prime between the k-th and (k+1)-th prime powers (A246655).
%C The prime powers themselves are: 3, 4, 7, 9, 13, 16, 23, 27, 31, 32, 47, 49, 61, 64, ...
%F A246655(a(n)) = A379157(n).
%e The 4th and 5th prime powers are 5 and 7, with interval (5,6,7) containing two primes, so 4 is not in the sequence.
%e The 13th and 14th prime powers are 23 and 25, with interval (23,24,25) containing only one prime, so 13 is in the sequence.
%e The 18th and 19th prime powers are 32 and 37, with interval (32,33,34,35,36,37) containing just one prime 37, so 18 is in the sequence.
%t v=Select[Range[100],PrimePowerQ];
%t Select[Range[Length[v]-1],Length[Select[Range[v[[#]],v[[#+1]]],PrimeQ]]==1&]
%Y These are the positions of 1 in A366835, for perfect powers A080769.
%Y For perfect powers instead of prime powers we have A378368.
%Y For no primes we have A379156, for perfect powers A274605.
%Y The prime powers themselves are A379157, for previous A175106.
%Y A000015 gives the least prime power >= n.
%Y A000040 lists the primes, differences A001223.
%Y A000961 lists the powers of primes, differences A057820.
%Y A031218 gives the greatest prime power <= n.
%Y A065514 gives the greatest prime power < prime(n), difference A377289.
%Y A246655 lists the prime powers.
%Y A366833 counts prime powers between primes, see A053607, A304521.
%Y Cf. A025474, A067871, A068315, A080101, A178700, A345531, A377281, A377283, A377287, A377434, A378374.
%K nonn,new
%O 1,1
%A _Gus Wiseman_, Dec 22 2024